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"... Abstract. We study set-theoretic solutions (X, r) of the Yang-Baxter equations on a set X in terms of the induced left and right actions of X on itself. We give a characterization of involutive square-free solutions in terms of cyclicity conditions. We characterise general solutions in terms of an i ..."

Abstract. We study set-theoretic solutions (X, r) of the Yang-Baxter equations on a set X in terms of the induced left and right actions of X on itself. We give a characterization of involutive square-free solutions in terms of cyclicity conditions. We characterise general solutions in terms of an induced matched pair of unital semigroups S(X, r) and construct (S, rS) from the matched pair. Finally, we study extensions of solutions in terms of matched pairs of their associated semigroups. We also prove several general results about matched pairs of unital semigroups of the required type, including iterated products S ⊲ ⊳ S ⊲ ⊳ S underlying the proof that rS is a solution, and extensions (S ⊲ ⊳ T, rS⊲⊳T). Examples include a general ‘double ’ construction (S ⊲ ⊳ S, rS⊲⊳S) and some concrete extensions, their actions and graphs based on small sets. 1.

...me that r is non-degenerate, as will be indicated. Also, as a notational tool, we shall often identify the sets X × X and X 2 , the set of all monomials of length two in the free semigroup 〈X〉. As in =-=[8]-=- to each quadratic map r : X 2 → X 2 we associate canonically algebraic objects (see Definition 2.2) generated by X and with quadratic defining relations ℜ naturally determined as (1.3) ℜ = ℜ(r) = {(u...

"... Abstract. We study finite set-theoretic solutions (X, r) of the Yang-Baxter equation of square-free multipermutation type. We show that each such solution over C with multipermutation level two can be put in diagonal form with the associated Yang-Baxter algebra A(C, X, r) having a q-commutation form ..."

Abstract. We study finite set-theoretic solutions (X, r) of the Yang-Baxter equation of square-free multipermutation type. We show that each such solution over C with multipermutation level two can be put in diagonal form with the associated Yang-Baxter algebra A(C, X, r) having a q-commutation form of relations determined by complex phase factors. These complex factors are roots of unity and all roots of a prescribed form appear as determined by the representation theory of the finite abelian group G of left actions on X. We study the structure of A(C, X, r) and show that they have a •-product form ‘quantizing ’ the commutative algebra of polynomials in |X | variables. We obtain the •-product both as a Drinfeld cotwist for a certain canonical 2-cocycle and as a braided-opposite product for a certain crossed G-module (over any field k). We provide first steps in the noncommutative differential geometry of A(k, X, r) arising from these results. As a byproduct of our work we find that every such level 2 solution (X, r) factorises as r = f ◦ τ ◦ f −1 where τ is the flip map and (X, f) is another solution coming from X as a crossed G-set. 1.

...ion G ⊆ ∏ 1≤i≤t Gi. The notions of retraction of symmetric sets and multipermutation solutions were introduced in the general case in [ESS], where (X, r) is not necessarily finite, or square-free. In =-=[GI]-=-, [GIM1], [GIM2] are studied especially the multipermutation square-free solutions of finite order, we recall some notions and results. Let (X, r) be a nondegenerate symmetric set. An equivalence rela...

"... A review of some recent results on the dynamical theory of the Yang-Baxter maps (also known as set-theoretical solutions to the quantum Yang-Baxter equation) is given. The central question is the integrability of the transfer dynamics. The relations with matrix factorisations, matrix KdV solitons, ..."

A review of some recent results on the dynamical theory of the Yang-Baxter maps (also known as set-theoretical solutions to the quantum Yang-Baxter equation) is given. The central question is the integrability of the transfer dynamics. The relations with matrix factorisations, matrix KdV solitons, Poisson Lie groups, geometric crystals and tropical combinatorics are discussed and demonstrated on several concrete examples.

"... We show the intimate connection between various mathematical notions that are currently under active investigation: a class of Garside monoids, with a “nice” Garside element, certain monoids S with quadratic relations, whose monoidal algebra A = kS has a Frobenius Koszul dual A! with regular socle, ..."

We show the intimate connection between various mathematical notions that are currently under active investigation: a class of Garside monoids, with a “nice” Garside element, certain monoids S with quadratic relations, whose monoidal algebra A = kS has a Frobenius Koszul dual A! with regular socle, the monoids of skew-polynomial type (or equivalently, binomial skew-polynomial rings) which were introduced and studied by the author and in 1995 provided a new class of Noetherian Artin-Schelter regular domains, and the square-free set-theoretic solutions of the Yang-Baxter equation. There is a beautiful symmetry in these objects due to their nice combinatorial and algebraic properties.

...isfies l1. (3) (X, r) satisfies r1. (4) (X, r) satisfies lr3 In this case (X, r) is cyclic and satisfies lri. Several other characterizations are already known in the case when X is finite, see [23], =-=[16]-=-, [15] from where we extract the following. Facts 2.10. Suppose that (X, r) is a finite quantum binomial set, | X |= n, let A = A(k, X, r) be the associated quadratic algebra over a field k. Then any ...

... 2012, 1s34 solutions of the Yang-Baxter equation attracts the attention of a broad circle of scientists (includingsmathematicians). Some suggested references related to our paper could be References =-=[4,5,6,7,8]-=-, etc.sIn this paper we present qualitative results concerning the (set-theoretical) Yang-Baxter equation.sWe first consider the solutions arising from relations. For any relation on a given set we co...

By means of left quasigroups L = (L, ·) and ternary systems, we construct dynamical Yang-Baxter maps associated with L, L, and (·) satisfying an invariance condition that the binary operation (·) of the left quasigroup L defines. Conversely, this construction characterize such dynamical Yang-Baxter maps. The unitary condition of the dynamical Yang-Baxter map is discussed. Moreover, we establish a correspondence between two dynamical Yang-Baxter maps constructed in this paper. This correspondence produces a version of the vertex-IRF correspondence.

"... Let k be a field and X be a set of n elements. We introduce and study a class of quadratic k-algebras called quantum binomial algebras. Our main result shows that such an algebra A defines a solution of the classical Yang-Baxter equation (YBE), if and only if its Koszul dual A! is Frobenius of dime ..."

Let k be a field and X be a set of n elements. We introduce and study a class of quadratic k-algebras called quantum binomial algebras. Our main result shows that such an algebra A defines a solution of the classical Yang-Baxter equation (YBE), if and only if its Koszul dual A! is Frobenius of dimension n, with a regular socle and for each x, y ∈ X an equality of the type xyy = αzzt, where α ∈ k \ {0}, and z, t ∈ X is satisfied in A. We prove the equivalence of the notions a binomial skew polynomial ring and a binomial solution of YBE. This implies that the Yang-Baxter algebra of such a solution is of Poincaré-Birkhoff-Witt type, and possesses a number of other nice properties such as being Koszul, Noetherian, and an Artin-Schelter regular domain.

... which contains various algebras, such as binomial skew polynomial rings, [9], [10], [11], the Yang-Baxter algebras defined via the so called binomial solutions of the classical Yang-Baxter equation, =-=[14]-=-, the semigroup algebras of semigroups of skew type, [15], etc. all of which are actively studied. Definition 1.2. Let A(k, X, ℜ) = k〈X〉/(ℜ) be a finitely presented k-algebra with a set of generators ...

"... Abstract. Set-theoretic solutions of the Yang–Baxter equation form a meetingground of mathematical physics, algebra and combinatorics. Such a solution consists of a set X and a function r: X ×X → X ×X which satisfies the braid relation. We examine solutions here mainly from the point of view of fini ..."

Abstract. Set-theoretic solutions of the Yang–Baxter equation form a meetingground of mathematical physics, algebra and combinatorics. Such a solution consists of a set X and a function r: X ×X → X ×X which satisfies the braid relation. We examine solutions here mainly from the point of view of finite permutation groups: a solution gives rise to a map from X to the symmetric group Sym(X) on X satisfying certain conditions. Our results include many new constructions based on strong twisted union and wreath product, with an investigation of retracts and the multipermutation level and the solvable length of the groups defined by the solutions; and new results about decompositions and factorisations of the groups defined by invariant subsets of the solution. Contents

"... Abstract. We study quadratic algebras over a field k. We show that an n-generated PBW algebra A has finite global dimension and polynomial growth iff its Hilbert series is HA(z) = 1/(1−z)n. Surprising amount can be said when the algebra A has quantum binomial relations, that is the defining relatio ..."

Abstract. We study quadratic algebras over a field k. We show that an n-generated PBW algebra A has finite global dimension and polynomial growth iff its Hilbert series is HA(z) = 1/(1−z)n. Surprising amount can be said when the algebra A has quantum binomial relations, that is the defining relations are nondegenerate square-free binomials xy − cxyzt with non-zero coefficients cxy ∈ k. In this case various good algebraic and homological properties are closely related. The main result shows that for an n-generated quantum bino-mial algebra A the following conditions are equivalent: (i) A is a PBW algebra with finite global dimension; (ii) A is PBW and has polynomial growth; (iii) A is an Artin-Schelter regular PBW algebra; (iv) A is a Yang-Baxter algebra; (v) HA(z) = 1/(1 − z)n; (vi) The dual A! is a quantum Grassman algebra; (vii) A is a binomial skew polynomial ring. So for quantum binomial alge-bras the problem of classification of Artin-Schelter regular PBW algebras of global dimension n is equivalent to the classification of square-free set-theoretic solutions of the Yang-Baxter equation (X, r), on sets X of order n. 1.